Formula (5) is generally speaking, an approximate equation, but if f(x) is a polynomial of degree not higher thann - 1, then the equation will be exact. This circumstance is what permits determining the quantities \(C_{n}, x_{1}, x_{2}, ..., x_{n} \).

To obtain a formula that is convenient for any interval of integration, let us transform the interval of integration [a,b] into the interval [-1, 1]. To do this, put
\[x = \frac{a+b}{2} + \frac{b-a}{2}t\]
then for t = -1 we will have x = a, for t=1, x=b.

where \(\Phi(t)\) denotes the function of t under the integral sign. Thus, the problem of integrating the given function f(x) on the interval [a,b] can always be reduced to integrating some other function \(\Phi(x)\) on the interval [-1,1].

From these n-1 equations we find the abscissas \(x_{1}, x_{2}, ..., x_{n} \). These solutions were found by Chebyshev for various values of n. The following solutions are those that he found for cases when the number of intermediate points N is equal to 3, 4, 5, 6, 7, 9:

Thus, on the interval [-1,1], an integral can be approximated by the following Chebyshev formula:
\[\int_{-1}^{1}{f(x)dx}= \frac{2}{n}[f(x_{1}) + f(x_{2}) + ... + f(x_{n})]\]
where n is one of the numbers 3, 4, 5, 6, 7, 9, and \(x_{1}, ..., x_{n}\)are the numbers given in the table. Here, n cannot be 8 or any number exceeding 9, for then the system of equations (10) yields imaginary roots.

When the given integral has limits of integration a and b, the Chebyshev formula takes the form
\[\int_{a}^{b}{f(x)dx}=\frac{b-a}{n}[f(X_{1}) + f(X_{2}) + ... + f(X_{n})]\]

where \(X_{i}=\frac{b+a}{2} + \frac{b-a}{2}x_{i}\) {i=1, 2, ..., n) and \(x_{I}\) have the values given in the table.

Example: Evaluate \(\int_{1}^{2}{\frac{dx}{x}}=ln(2)\)

Solution: First, by a change of variable, transform this integral into a new one with limits of integration -1 to 1:
\[x+\frac{1+2}{2}+\frac{2-1}{2}t = \frac{3}{2} + \frac{t}{2} = \frac{3+t}{2}\]
\[dx = \frac{dt}{2}\]